Umm yeah the question asks does blah=blah. I thought that it would not be true, however I am clueless how to put it on paper that the direction would not be the same. Should I call sum of vectors b and c a vector d?

Generally, the easier the counterexample the better. Since the reason for the deficit is known it can help you to find an example:
Take b and c linearly dependent. Then projecting on b is the same as projecting on c or projecting on b+c. So the right side is twice the left side.